Nonlinear Systems Stability Analysis

Appendix A5: Some New Notions for Symmentric Behavior of Matrices and Related Theorems

Throughout this appendix, it is assumed that $\mathbb{F}$ is an arbitrary field. Let m,n e N, we denote the ring of all n× n matrices and the set of all m× n matrices over $\mathbb{F}$ by $M_n\left(\mathbb{F}\right)$ and $M_{m\times n}\left(\mathbb{F}\right)$ respectively, and for simplicity let ${\mathbb{F}}^n=M_{1\times n}\left(\mathbb{F}\right)$ The zero matrix is denoted by 0. Also the element of an m× n matrix, R on the i-th row and j-th column is denoted by r(ij). If $A=\left(a_{ij}\right)\in M_n\left(\mathbb{F}\right)$ then ${\left[A\right]}_{ij}\in M_{\left(n-1\right)}\left(\mathbb{F}\right)$ is a matrix obtained from omitting the i-th row and the j-th column of the matrix A Also, $\text{cof}\left(A\right)\in M_n\left(\mathbb{F}\right)$ is defined by cof(A) = (a_ij(–1)^i+jdet([A]_ij). For any ij1 ≤ ij ≤ n, we denote by E_ij that element in $M_n\left(\mathbb{F}\right)$ whose (ij) entry is 1 and whose other entries are 0. Also, 0 ...